Published online by Cambridge University Press: 10 May 2019
We consider a spherical variant of the Faraday problem, in which a spherical drop is subjected to a time-periodic body force, as well as surface tension. We use a full three-dimensional parallel front-tracking code to calculate the interface motion of the parametrically forced oscillating viscous drop, as well as the velocity field inside and outside the drop. Forcing frequencies are chosen so as to excite spherical harmonic wavenumbers ranging from 1 to 6. We excite gravity waves for wavenumbers 1 and 2 and observe translational and oblate–prolate oscillation, respectively. For wavenumbers 3 to 6, we excite capillary waves and observe patterns analogous to the Platonic solids. For low viscosity, both subharmonic and harmonic responses are accessible. The patterns arising in each case are interpreted in the context of the theory of pattern formation with spherical symmetry.
Visualisation of $\ell=1$ mode for spherical drop. The drop is displaced alternately to the left and the right. Length and colors of arrows indicate the velocity of the surrounding air.
Visualisation of $\ell=2$ prolate-oblate pattern of gravitational harmonic waves. Drop interface and velocity field on and outside drop are shown.
Visualisation of $\ell=3$ tetrahedral pattern of capillary subharmonic waves.
Visualisation of $\ell=4$ cubic-octahedral pattern of capillary subharmonic waves.
Visualisation of $\ell=4$ axisymmetric pattern for subharmonic capillary waves. In this stroboscopic film, only snapshots at a single temporal phase are included, emphasizing the overall drift of the pattern
Visualisation of $\ell=5$ mode for subharmonic capillary waves, showing evolution from axisymmetric to $D_4$ pattern.